Dynamic asset allocation using option implied distributions in an exponentially tempered stable Lévy market

(1)

M

ASTER

M

ATHEMATICAL

F

INANCE

M

ASTER’S

F

INAL

W

ORK

D

ISSERTATION

D

YNAMIC

A

SSET

A

LLOCATION

U

SING

O

PTION

I

MPLIED

D

ISTRIBUTIONS

I

N

A

N

E

XPONENTIALLY

T

EMPERED

S

TABLE

L

ÉVY

M

ARKET

(2)

M

ASTER

M

ATHEMATICAL

F

INANCE

M

ASTER’S

F

INAL

W

ORK

D

ISSERTATION

D

YNAMIC

A

SSET

A

LLOCATION

U

SING

O

PTION

I

MPLIED

D

ISTRIBUTIONS

I

N

A

N

E

XPONENTIALLY

T

EMPERED

S

TABLE

L

ÉVY

M

ARKET

Z

ACHARY

P

OLASKI

S

UPERVISORS:

J

OÃO

G

UERRA

M

ANUEL

G

UERRA

S

EPTEMBER -

2018

(3)

Dynamic Asset Allocation Using Option-Implied Distributions in

an Exponentially Tempered Stable Lévy Market

Zachary Polaski

Submitted in partial fulfillment of the requirements for the degree of Masters of Science

(Mathematical Finance)

at the

University of Lisbon School of Economics and Management (ISEG) 2018

Abstract:

This paper explores the optimal portfolio problem using option-implied distributions when the underlying price process is assumed to be driven by an exponential Lévy process. In particular, the application is carried out using an Exponentially Tempered Stable jump-diffusion process as the martingale component of the log stock price, and the investor’s preferences are assumed subject to a CRRA utility function. One month risk-neutral densities are extracted from option prices by using a transform pricing method and are subsequently transformed to the risk-adjusted, or real-world density via a model preserving minimal entropy transform which importantly maintains the parameterization of the Lévy process. A stochastic optimal control result is then used to construct a portfolio consisting of a risky and risk-free asset which is rebalanced on a monthly basis. It is found that the portfolios formed using option-implied expectations under the Lévy market assumption, which are flexible enough to capture the higher moments of the implied distribution, are far more robust to left-tail market risks and offer statistically significant improvements to risk-adjusted performance when investor risk aversion is low, however this diminishes as risk aversion increases.

(4)

Resumo:

Este artigo explora o problema do portfólio ideal usando distribuições implícitas na opção quando o processo de preço subjacente é assumido como sendo conduzido por um processo exponencial de Lévy. Em particular, a aplicação é levada a cabo usando um processo de difusão de salto Estável Exponencialmente Temperado como o componente martingale do preço das acções de log, e as preferências do investidor são assumidas sujeitas a uma função de utilidade CRRA. Densidades de um mês neutras ao risco são extraídas dos preços das opções usando um método de precificação por transformação e são subsequentemente transformadas na densidade ajustada ao risco ou no mundo real por meio de um modelo preservando a entropia mínima que mantém a parametrização do processo Lévy. Um resultado de controle otimizado estocástico é então usado para construir um portfólio que consiste em um ativo de risco e sem risco, que é reequilibrado mensalmente. Descobriu-se que os portfólios formados usando as expectativas implícitas na opção sob a hipótese de mercado Lévy, que são flexíveis o suficiente para capturar os momentos mais altos da distribuição implícita, são muito mais robustos aos riscos de cauda esquerda e oferecem melhorias estatisticamente significativas ao desempenho ajustado ao risco quando a aversão ao risco do investidor é baixa, porém isso diminui à medida que aumenta a aversão ao risco.

(5)

1 Introduction

1.1 Motivations

This paper undertakes a study of the fundamental optimal asset allocation problem in finance and extends the current literature by assessing the (simulated) empirical performance of a dynamic strategy which relies on signals generated from extracting information from option prices under an exponential Lévy market assumption. The paper builds mainly off of the results and methodology of [37], which showed that portfolio performance could be improved by using implied forward-looking rather than historical distributions, and [32] and [58], where a number of results were derived for the optimal control strategy of exponential Lévy processes. Given that the parameterizations of Lévy processes used in financial modeling (including in particular the Exponentially Tempered Stable process used throughout this study) have economic interpretations and are supposed to offer improvements over the more simple classical models, forward-looking estimates of the parameters could reasonably be expected to improve realized portfolio performance within a framework which incorporates this information. The goal is thus to investigate whether a number of rather esoteric results in stochastic processes, option pricing, and optimal stochastic control together have any value in a quite simple and implementable investment strategy. In essence, the objective of the research was to replicate the broader result of [37], but whereas there the authors used a non-parametric approach to extract the implied distributions, here a parametric approach (i.e. an option pricing model) is used to model the distribution implied in option prices.

1.2 Literature Review

The extensive literature regarding multi-period portfolio selection was catalyzed by the seminal papers of Mossin [54], Samuelson [63], and Merton [51] in the 1960s. By applying the ideas of dynamic programming and stochastic optimal control to the portfolio selection problem, with these papers emerged an exchange of fundamental concepts in engineering to navigating the financial decision making process. But perhaps it was the widely known paper of Fischer Black and Myron Scholes, "The Pricing of Options and Corporate Liabilities" [8], that truly kicked off the love affair of finance and mathematics. The shortcomings of the Black-Scholes model, however, became all too apparent, particularly in the aftermath of the events of October 19th, 1987, now known as "Black Monday," which saw a permanent regime change in the implied volatility structure that the Black-Scholes model simply could not handle. In [46] much of the current framework for using Lévy processes for price modeling was first developed with the Variance Gamma (VG) process, although the idea of modeling prices with infinite activity and discontinuities had already been considered years earlier in [49], [60], [59], and [9]. Since the Variance Gamma process, a number of additional Lévy processes have been proposed to more realistically capture stylized facts of the markets, perhaps the most popular being the Hyperbolic process of [22], the Normal Inverse Gaussian process of [3], the CGMY process of [12], the Kou process of [38], and the Meixner process of [67]. This study involves the merging of the optimal portfolio selection literature and the literature on Lévy processes in finance within a systematic investment strategy. The problem of optimizing a portfolio with Lévy drivers was first solved using the dynamic programming approach in [6] and [24].

(8)

In [32], the problem was also solved using the alternative martingale method approach, and this remains a seminal reference herein due to the clarity of the notation and results (the solutions are identical, as they should be, between the two approaches). As mentioned in the introduction, more recently, in [58], the optimal portfolio problem was studied in more depth numerically for exponential Lévy processes under CRRA utility. However, this paper only considered the problem cross-sectionally, that is it compared optimal portfolios under the jump assumption to the pure diffusion assumption using a single Lévy process parameterization/calibration and studied the implications. We pick up where this study left off, inspired, as mentioned above, by [37] and analyze the problem across time and investigate the performance of a periodically rebalanced strategy.

1.3 Organization of Paper

The rest of the paper is organized as follows. Section 2 is a review of foundational definitions and theorems important in formulating and understanding the results of later sections. Section 3 details the Exponentially Tempered Stable Lévy process, including the application to financial markets and an important result regarding transformations between the risk-neutral and real-world parameterization. Section 4 derives a suitable Hamilton-Jacobi-Bellman (HJB) equation for the optimal investment problem, and is solved assuming the solution for the portfolio proportions lies within the interior of a set of admissible controls and the investor’s preferences are subject to the logarithmic and power HARA-type utility functions. Section 5 is a review of the Carr-Madan transform option pricing methodology. Section 6 presents the main results of the paper, where the topics of previous sections are brought together to create and backtest an investable strategy involving a simple portfolio consisting of a risky and risk-free asset rebalanced on a monthly basis.

2 Lévy Processes

2.1 Basic Definitions and Theorems

This chapter presents a number of foundational definitions and theorems regarding Lévy processes. The intent is for this text to be sufficiently self-contained that one need not consult a myriad of external sources. Nevertheless, a full treatment of the theory could never be condensed (nor is it at all the point of the paper) to a few pages, and should the reader desire a more extensive, yet largely introductory treatment, [1], [19], and [66] are recommended. What follows is mainly sourced from these texts. We begin by simply defining a Lévy process.

Definition 2.1 (Lévy Process) A real valued and adapted stochastic process L = {Lt, t ≥ 0} defined on

the probability space(Ω, F , P) is called a Lévy process if it possesses the following properties: 1.L0 = 0 (a.s).

2. L has independent and stationary increments, i.e., Lt− Ls is independent ofFs for anys < t and

Lt− Lshas the same law asLt−s.

(9)

More simply, a Lévy process can be thought of as the continuous-time analog of a random walk. Every Lévy process can be associated with an infinitely divisible distribution, defined as follows.

Definition 2.2 (Infinitely Divisible Distribution) The distribution of a random variable X is infinitely divisible if and only if for all_{n ∈ N there exists a random variable X}(1/n)such that

φX(u) = (φ_X(1/n)(u))n, (2.1)

whereφX(u) is the characteristic function of X and φX(1/n) is the characteristic function ofX(1/n).

Equivalently, the probability distribution is infinitely divisible if there exists a convolutionn-th root for

each_{n ∈ N.}

A necessary and sufficient condition for an infinitely divisible distribution is the Lévy Khintchine formula, which generalizes the characteristic function for all Lévy processes as follows.

Theorem 2.3 (Lévy-Khintchine Representation) A random variable is infinitely divisible if and only if there exists a triplet(µ, σ2_{, ν) where µ ∈ R, σ ∈ R}+

0 andν is a positive sigma-finite measure on R, such

that the characteristic function is given by

φX(u) = eψ(u), (2.2)

whereψ(u) is the characteristic exponent, given by the expression

ψ(u) = iµu − 1 2σ 2_u2₊ Z R (eiux− 1 − iux

χ

_{{|x|≤1}(x))ν(dx),} (2.3)

and

χ

_{A(x) denotes the indicator function, defined as}

χ

A(x) :=

(

1 if x ∈ A,

0 if x /∈ A. (2.4)

Since every Lévy process is associated to an infinitely divisible distribution, the characteristic function can also be expressed as

φLt(u) = e

tψ(u)_, _(2.5)

where L = {Lt, t ≥ 0} is the Lévy process associated with the triplet (µ, σ2, ν). The positive measure

ν is referred to as the Lévy measure of L, which must satisfy certain properties given in the following definition.

Definition 2.4 (Lévy Measure, Paths, and Moment Properties) The Lévy measure ofL, denoted ν, must satisfy the following properties:

(

1.) ν{0} = 0

2.) R

R(1 ∧ x

2_{)ν(dx) < ∞, where A ∧ B := min(A, B)} . (2.6)

The Lévy measure gives the expected number of jumps of a certain size per unit of time. The following properties describe a Lévy process and its measure on a given bounded nontrivial interval:

(10)

• If ν(R) = ∞ then infinitely many small jumps occur. The Lévy process is said to have infinite

activity. (2.7)

• If ν(R) < ∞ then a.a. paths have a finite number of jumps. The Lévy process is said to have finite

activity. (2.8)

• Let L be a Lévy process with Lévy triplet (µ, σ2_{, ν). If σ}2 _{= 0 and}R

|x|≤1|x|ν(dx) < ∞, then

a.a. paths have finite variation. If σ2 6= 0 orR

|x|≤1|x|ν(dx) = ∞, then a.a. paths have infinite

variation. (2.9)

While the path variation properties above are seen to be only related to the small jumps and/or the Brownian motion components, the moment properties depend on the large jumps in the following way:

• L_thas finite moment of order p if and only if Z

|x|≥1

|x|pν(dx) < ∞. (2.10)

• L_t_{has finite exponential moment of order p (i.e. E[e}pLt_{] < ∞) if and only if}

Z

|x|≥1

epxν(dx) < ∞. (2.11)

The class of stable distributions is an important subclass of infinitely divisible distributions. As the name suggests, the Lévy process we will focus on later in this paper, the Exponentially Tempered Stable process, incorporates exponential decay or "tempering" terms into the Lévy measure of an α-stable random variable, presented in the following theorem (for proof, see [64]).

Theorem 2.5 (Lévy Measure of α−Stable Random Variable) If X is a stable random variable, that is there exists sequences{cn, n ∈ N}, {dn, n ∈ N}, cn> 0 such that a linear combination of independent

copiesX1+ ... + Xn= cd nX + dn, then 1. W hen α = 2, X ∼ N (µ, A). 2. W hen α 6= 2, A = 0, and ν(dx) = ( _c 1 x1+αdx if x > 0 c2 |x|1+αdx if x < 0 , where c1, c2 ≥ 0 and c1+ c2 > 0. (2.12)

It can be shown that if X is stable, then cn = σn

1

α with α a constant in the interval (0, 2]. The

parameter α is called the stability index and is meaningful in characterizing the tails of the distribution. An important feature of stable laws is the existence of polynomial or "fat" tails when α 6= 2. Thus, for α < 2, the distribution has an infinite variance, and furthermore has an undefined mean or expected value for α < 1. As shown above, the case when α = 2 is the normal distribution, and thus the tails decay exponentially and do not exhibit "fatness."

(11)

An important structural property of Lévy densities is monotonicity, and we will require that the Lévy processes we work with have completely monotone jumps. Recall that a function f : (0, ∞) 7→ R is said to be completely monotone if it possesses derivatives f(n)(x) and (−1)nf(n)(x) ≥ 0 for all n = 0, 1, 2... and x > 0. That is, derivatives of the same order have the same sign and are alternating in sign. A completely monotone Lévy density structurally relates arrival rates of large jump sizes to smaller jump sizes. Intuitively, we should expect that large jumps occur less frequently than small jumps. A well known necessary and sufficient condition for a function to be completely monotone is given in the following theorem.

Theorem 2.6 (Bernstein’s Theorem) A function f is completely monotone if and only if it is the Laplace transform of some non-negative measureν,

f (x) =

Z ∞

0

e−axν(da). (2.13)

Put another way, complete monotonicity asserts that for differentiable densities the derivative is negative for positive jumps and positive for negative jumps. This leads us to the following definition, taken from [29].

Definition 2.7 (Completely Monotone Jumps) The process X has completely monotone jumps if its Lévy measure is absolutely continuous with densityf decreasing exponentially when x → ±∞ and such thatf (x) and f (−x) are completely monotone.

The concept of completely monotone jumps is perhaps best grasped by a visual inspection of the density of a Lévy measure. The figures below, from [31], plot the Lévy measures of two processes known as the Variance Gamma (left) and the Linear Gamma (right) processes with varying parameters (the specifics of these processes are not important). The size of the jump, x, corresponds to the horizontal axis and the number of jumps corresponds to the vertical axis. One sees that for negative jumps, we have monotonically increasing functions with a positive derivative and relatively more small jumps than large, and for positive jumps, monotonically decreasing functions with a negative derivative, but also with relatively more small jumps than large.

(12)

Finally, since many, if not most, Lévy processes do not have explicitly known probability densities, we are in general confined to work with the characteristic function, so it is helpful to recall the relationship between these functions.

Theorem 2.8 (Probability Density and Characteristic Function Bijection) Let φX(u) be the characteristic

function of some real-valued random variable. Then, there exists a bijection between the probability distribution and the characteristic function. That is, two distinct probability distributions never share the same characteristic function. Since the characteristic function is the Fourier transform, denotedF , of the lawf ,

φX(u) = F [f ](u) =

Z ∞

−∞

eiuxf dx, (2.14)

the probability density function can be recovered by a Fourier inversion of the characteristic function, f (x) = F−1[φX(u)](x) = 1 2π Z ∞ −∞ e−iuxφX(u)du, (2.15)

whereφX(u) denotes the characteristic function of x evaluated at u.

2.2 Itô Calculus for Lévy Processes

Theorem 2.9 (The 1-dimensional Itô Formula) Suppose Xtis a Lévy-type stochastic integral of the

form dXt= µtdt + σtdWt+ Z R g(t, z) ¯N (dt, dz), (2.16) where ¯ N (dt, dz) = ( N (dt, dz) − ν(dz)dt if|z|< R N (dt, dz) if|z|≥ R, (2.17)

for someR ∈ [0, ∞] and Wtis a standard Wiener process.

Letf ∈ C2(R2) and define Yt= f (t, Xt). Then Ytis again a Lévy-type stochastic integral and

dYt= ∂f ∂t(t, Xt)dt + ∂f ∂x(t, Xt)[µtdt + σtdWt] + 1 2σ 2 t ∂2f ∂x2(t, Xt)dt + Z |z|<R {f (t, X_t−+ g(t, z)) − f (t, X_t−) − g(t, z)∂f ∂x(t, Xt−)}ν(dz)dt + Z R {f (t, X_t−+ g(t, z)) − f (t, X_t−)} ¯N (dt, dz), (2.18) and ( IfR = 0, then ¯N = N everywhere IfR = ∞, then ¯N = ˜N everywhere.

N is the so-called Poisson random measure, or a family of Poisson distributed random variables, with ˜N being the centered or "compensated" Poisson random measure in which the additional term is the quantity which must be subtracted from N in order to obtain a martingale. A more comprehensive treatment of these random measures can be found in [19].

(13)

2.3 Stochastic Exponential Lévy Processes

Consider a one-dimensional process L = {Lt, t ≥ 0}, which is a solution of the stochastic differential

equation (SDE)

dLt= Lt−dX_t, (2.19)

where X is a Lévy-type stochastic integral of the form dXt= µtdt + σtdWt+ Z |z|<1 g(t, z) ˜N (dt, dz) + Z |z|≥1 w(t, z)N (dt, dz), (2.20)

where a tilde denotes a compensated Poisson integral (importantly a martingale), which has the decomposition Z A g(t, z) ˜N (dt, dz) = Z A g(t, z)N (dt, dz) − Z A g(t, z)ν(dz)dt. (2.21)

The solution of (2.19) is the "stochastic exponential" or the "Doléans-Dade exponential," Lt= EX(t) = exp{Xt− 1 2σ 2 t}

Π

0≤s≤t(1 + 4Xs)e −4Xs_. _(2.22)

For financial applications, we will require that (to avoid negative prices)

inf{4Xt, t ≥ 0} ≥ −1 a.s. (2.23)

An alternative form for (2.22) is

EX(t) = eSX(t), (2.24) where dSX(t) = (µt− 1 2σ 2 t)dt + σtdWt+ Z |z|≥1 log(1 + w(t, z))N (dt, dz) + Z |z|<1 log(1 + g(t, z)) ˜N (dt, dz) + Z |z|<1 (log(1 + g(t, z)) − g(t, z))ν(dz)dt. (2.25)

2.3.1 Relation Between Ordinary and Stochastic Exponentials

An important relation exists between stochastic and ordinary exponentials. Lévy models and their associated triplets are usually presented in the literature assuming ordinary exponential form, i.e. St =

S0eLtfor the stock price. However, as we have just seen the popular geometric type stochastic differential

of equation (2.19) leads to a stochastic exponential type solution. Fortunately, there is a clear relation between these formulations, given by the following theorem from [19]:

Theorem 2.10 If L is a Lévy process with characteristic triplet (µ, σ2, ν), then EL(t) = eL1(t)whereL1

is a Lévy process with characteristic triplet(µ1, σ12, ν1), given by

(14)

µ1 = µ − 1₂σ2+

R

R[log(1 + x) − x]ν(dx),

σ1= σ. (2.26)

Conversely, there exists a Lévy processL2 with characteristic triplet(µ2, σ22, ν2) such that there exists

eLt _{= E} L2(t), where ν2= ν ◦ g−1, g(x) = ex− 1, µ2 = µ + 1₂σ2+ R R[(e x_{− 1) − x]ν(dx),} σ2= σ. (2.27)

2.4 Exponential Lévy Stock Price Processes, EMMs, and Esscher Transforms

As was just briefly noted, exponential Lévy process can be suitable models for stock prices, given by

St= S0eLt, (2.28)

where

Lt= Xt+ µt. (2.29)

That is, the Lévy process X is taken as the martingale component of the log-stock price process. However, the Lévy process itself will have an associated drift, and therefore we must make an appropriate adjustment. Rather than evaluating each Lévy process individually and compensating for the drift, the martingale condition can be satisfied by compensating for

E[eLt], (2.30)

which is simply the characteristic function evaluated at −i. Accordingly, we add a mean-correction term to the drift of the exponential price process, given by

ω = − log[φ(−i)], (2.31)

where φ(·) is the characteristic function of the Lévy process. The final form of the price process is thus given by

St= S0eLt+ωt. (2.32)

The mean-correcting measure can be employed such that the discounted price ˜St= e−rtStis a martingale.

Following [66], the procedure is:

1) Estimate the parameters of the process by some suitable method

2) Change the drift term of the old process, call it some µold, in such a way that

(15)

Exchanging µnewfor µoldin the Lévy process implies the discounted price ˜St= e−rtStis a martingale.

Still, it is important to note that this equivalent martingale measure is not unique in most cases involving Lévy processes. We must therefore have a criterion to reduce the class of possible measures Q to an

appropriate subset and then obtain a unique equivalent measure Qu. Consider the Radon-Nikodym

derivative

dQu

dP |Ft= Nu(t). (2.34)

The measure Qu is called the Esscher transform of P by the martingale Nu, given by

N_u(t) = exp(−uXt+ tψ(u)). (2.35)

Significant results related to the Exponentially Tempered Stable process will be given in the next section. The important take-away here is the intuitive notion that the Esscher transform is such that the measure Qu minimizes the relative entropy (or "Kullback-Leibler distance") H(Q|P) between the measures Q

and P. The relative entropy H(Q|P) is given by

H(Q|P) = EQ log dQ dP = EP dQ dPlog dQ dP , (2.36)

and thus the measure Quis such that

H(Qu|P) = min EQ log dQ dP = min EP dQ dPlog dQ dP . (2.37)

A full treatment of this concept and a proof of the minimization of the relative entropy by Qu can be

found in [16].

3 The Exponentially Tempered Stable (ETS) Processes

The Lévy process used throughout the paper is the Exponentially Tempered Stable (hereafter ETS) process. The ETS process ([36] and [10]) is in fact just the CGMY process popularized in [12] with a generalization of one parameter based on the sign of the random variable (equivalently, the CGMY process is a special case of the ETS process). In fact, most, if not all, of the results presented below are easily understood if one is familiar first with the CGMY process, and [12] is the main reference for the mathematics presented herein. Later, in for instance [39], another special case known as Bilateral Gamma was studied. The tempered stable process has found many uses outside the field of finance, such as dynamical systems and fluid dynamics ([68], [69]). It is a popular model of turbulence in the physical sciences, where it is known more commonly as Truncated Lévy Flight ([50]). In a rather interesting application, the process has been shown, in for instance [72] and [73], to explain foraging behavior, and in [70] it was shown that great white sharks abandon a Brownian motion hunting pattern in favor of a Lévy flight as nearby prey becomes scarce. This has led to what is known as the Lévy Flight Foraging Hypothesis, which states that, as in [73], "since Lévy flights optimize random searches, biological organisms must have therefore evolved to exploit Lévy flights." But alas, we must now continue with the banalities of the financial applications.

(16)

3.1 Parameters, Properties, and Characteristic Function

The ETS process is a pure jump Lévy process with parameters λ+, λ−, β+, β−and α, with Lévy triplet

(µ, σ2, ν) given by: µ = ( b +R_|x|≤1xν(dx) if α < 1 b −R_|x|>1xν(dx) − λ+αΓ(−α)(βα−1+ ) + −λ−αΓ(−α)(β−α−1) if α ∈ (1, 2), σ = 0, ν(x) = λ+ exp(−β+x) xα+1

χ

x>0 + λ− exp(−β−|x|) |x|α+1

χ

x<0. (3.1)

where λ+, λ−, β+, β−> 0 and α < 2. The condition α < 2 ensures that the Lévy density will integrate

x2around 0. As the name suggests, the ETS process and a stable process with stability index α ∈ (0, 2) have similar Lévy measures, but importantly the measure of the ETS process includes the additional exponential "tempering" factors. Because of this exponential tempering within the Lévy measure, the distribution has finite moments of all orders (see Definition 2.4), and the large jumps do not require truncation. The parameters of the ETS process play an important role in capturing various desirable aspects in a stochastic process with applications to financial problems. The parameters λ+ and λ−

can be seen as a measure of the overall level of activity of the process and, all else equal (and only considering movements greater than some arbitrarily chosen small value), the aggregate activity level may be calibrated through movements in λ+and λ−. The parameters β+and β−, respectively, control

the rate of exponential decay on the right and left of the Lévy density, allowing for the construction of a skewed distribution. For example, if β− < β+ the exponential tempering factor for negative values

of the random variable is inducing slower decay than the positive factor and thus we have a left-skewed distribution, which is consistent with the risk-neutral distribution typically implied from option prices. In the special case where β− = β+, the Lévy measure is symmetric, although non-normal distributions

can still be generated through the parameters λ+ and λ−, which provide control over the kurtosis of

the random variable. Finally, the parameter α describes the behavior of the Lévy density near zero, and is useful in characterizing the fine structure of the stochastic process. Note that the case when α = 0 is the (generalized) Variance Gamma process of [45]. The relationship of the moments of the ETS random variable to the parameters are perhaps most easily grasped by considering the closed form expressions, and these are therefore given below in Theorem 3.1. It is clear from these equations that the variance, skewness, and kurtosis of the ETS random variable are all positively related to λ+, λ−and α

and negatively related to β−and β+(more specifically, negative skewness is inversely related to β−and

positive skewness is inversely related to β+, and, all else equal, overall skewness is positively related to

β−− β+). Additionally, the main properties of the ETS process, that is, its monotonicity (recall Theorem

(2.6)), level of activity, and its variation may be defined by the value of parameter α, and this follows in Theorem 3.2 (for proof see [12]).

(17)

process, consideringt = 1, are given by (for Proof see Appendix A): Variance= λ+Γ(2 − α) 1 β2−α₊ + λ−Γ(2 − α) 1 β₋2−α , (3.2) Skewness= λ+Γ(3 − α) 1 β3−α₊ Variance3/2 − λ−Γ(3 − α) 1 β3−α₋ Variance3/2 , (3.3) Kurtosis= 3 + λ+Γ(4 − α) 1 β+4−α Variance2 + λ−Γ(4 − α) 1 β−4−α Variance2 . (3.4)

Theorem 3.2 (Process Properties and Ranges for the Parameter α) The ETS process • has a completely monotone Lévy density for α > −1;

• is a process of infinite activity for α > 0; and

• is a process of infinite variation for α > 1 (3.5)

Importantly, even though the ETS process as presented to this point is a pure jump process with no Brownian motion component, the infinite activity property afforded by setting α > 0 can still capture small market fluctuations in the absence of a Brownian motion. Still, an orthogonal diffusion component can be added to improve the fine structure of the stock price process, resulting in a so-called "jump-diffusion" process. Although the probability density function of the ETS process is not explicitly known, the characteristic function of the ETS density admits a rather simple form, and is given in the following definition. For ease of notation, the vector−→v = {λ+, λ−, β−, β+, α} will be used to denote the ETS

parameters for the rest of the paper.

Theorem 3.3 (ETS Characteristic Function) The characteristic function of the ETS random variable is given by (for Proof see Appendix B):

φET S(u, t;−→v ) = exp(tλ+Γ(−α)[(β+− iu)α− β+α] + tλ−Γ(−α)[(β−+ iu)α− β−α]). (3.6)

The relatively straightforward characteristic function of the ETS process lends the pricing problem well to solutions via methods which directly relate the characteristic function to the option price such as transform methods which will be elaborated in a later section.

3.1.1 Statistical (Real-World) and Risk Neutral Price Processes

The ETS model for the stock price process takes an ETS random variable XET S as the martingale

component of the log stock price, making the statistical price process

(18)

where µ is the mean rate of return on the stock and ω is the mean-correction term detailed in section 2.4, given here by

ω = −log(φET S(−i)) = −λ+· Γ(−α)[(β+− 1)α− β+α] − λ−· Γ(−α)[(β−+ 1)α− β−α]. (3.8)

We can also consider the following "extended" or jump-diffusion model which adds an orthogonal diffusion component,

XET Se(t;

−→_{v , η) = X}

ET S(t;−→v ) + ηWt, (3.9)

with Wta standard Wiener process independent of XET S. The price process of this extended model is

St= S0· exp[(µ + ω − η2/2)t + XET Se(t;

−→_{v , η)].} _(3.10)

Integral to subsequent sections, the characteristic function for the logarithm of the stock price under this extended model is given by (for Proof see Appendix C):

φln(St)(u, t) = E

P_[eiuln(St)_]

= exp(iu{ln(S0) + (µ + ω − η2/2)t}) · φET S(u;−→v ) · exp(−η2u2/2). (3.11)

As an important corollary to the statistical price process above, the risk-neutral price process, which will later be calibrated from option market data, replaces the asset-specific drift µ with the risk-free rate r, yielding

St= S0· exp[(r + ˜ω − ˜η2/2)t + XET Se(t; ˜

−→

v , ˜η)], (3.12)

with tildes denoting the parameters are now set under the risk-neutral measure. In this context, the characteristic function of log returns becomes, by the same logic that led to equation (3.11),

˜

φ_ln(S_t)(u, t) = EQ[eiuln(St)]

exp(iu{ln(S0) + (r + ˜ω − ˜η2/2)t}) · φET S(u; ˜−→v ) · exp(−˜η2u2/2), (3.13)

with ˜ ω = −log(φET S(−i)) = − ˜λ+· Γ(− ˜α)[( ˜β+− 1)α˜− ˜β+ ˜ α ] − ˜λ−· Γ(− ˜α)[( ˜β−+ 1)α˜− ˜β− ˜ α ]. (3.14)

3.1.2 Minimal Entropy Martingale Measure of the ETS Process

The Esscher transform concept introduced in section 2.4 can be specialized to the ETS process, affording an important result to a later part of this study. First, consider the following proposition from [35]: Proposition 3.4 Suppose Xt, 0 ≤ t ≤ T is the ETS process with parameters (−→v ) under P and the ETS

process with parameters( ˜−→v ) under Q. Then P and Q are equivalent for all t > 0 if and only if α = ˜α, λ+= ˜λ+andλ−= ˜λ−.

(19)

This proposition is further built upon in [34] as follows. Consider again the characteristic exponent of the ETS process,

ψET S(u;−→v ) := λ+Γ(−α)[(β+− iu)α− β+α] + λ−Γ(−α)[(β−+ iu)α− β−α)]. (3.15)

Along with Proposition 3.4, we have the following:

Theorem 3.5 (EMM Conditions for the ETS Model) Assume St, 0 ≤ t ≤ T is the real-world ETS

stock price process with parameters(−→v , µ) under the measure P, and with parameters (−→v , (r − q))˜ under the measure Q. Then Q is an EMM of P if and only if α = ˜α, λ+= ˜λ+, λ− = ˜λ−, and

r − q − ψET S(−i; λ+, λ−, ˜β−, ˜β+, α) = µ − ψET S(−i;−→v ). (3.16)

Choosing someθ, −β−< θ < β+, we have

r − q − ψET S(−i; λ+, λ−, β−+ θ, β+− θ, α) = µ − ψET S(−i;−→v ). (3.17)

However, this martingale measure is not yet unique, in the sense that it does not guarantee the minimal entropy. Accordingly, we recall the Esscher transform concept and consider a "model preserving minimal entropy martingale measure," so-called in [34], given by

H(Qu|P) = H(Q_β˜₋_{, ˜}_β₊|P) = min{H(Q|P)|Q ∈ EMM(P)}, (3.18)

where Q ∈ EMM(P) denotes that the EMM condition of equation (3.17) is satisfied, and the relative entropy of the ETS process can be expressed explicitly by (for Proof see Appendix D):

H(Q|P) = tλ+Γ(−α)((α − 1) ˜β+ α − αβ₊β˜+ α−1 + β₊α) + tλ−Γ(−α)((α − 1) ˜β− α − αβ−β˜− α−1 + β₋α). (3.19)

4 Stochastic Optimal Control of Exponential Lévy Processes

4.1 The Dynamic Programming Principle

The dynamic programming principle was greatly advanced by Richard Bellman and his groundbreaking work in the 1950s (see [4] and [5]), although the idea had already been foreshadowed in works such as [74] and [75]. Bellman’s breakthrough involved casting the value of a decision problem as an expectation of the initial value of some state variables (defined shortly) and the value of the choices to be made subsequently throughout the horizon of the decision problem. In a discrete time setting, a dynamic optimization problem is broken up into a sequence of discrete subproblems which are recursively optimized via backward induction, that is beginning with the last subsequence. The essential aspect of these problems, known as Bellman equations, is the necessary condition brought about by the Principle of Optimality, which roughly states,

(20)

Given an optimal sequence of decisions or choices, each subsequence must also be optimal. Considering the "decision problem" to be that of a one-dimensional portfolio strategy (i.e. a single risky asset and without consumption), the investor’s goal could be thought of as maximizing terminal or end-of-period wealth,

EP[U (XTπ)|Xt= x], (4.1)

where U (·) denotes some applicable investor utility function and P denotes the problem is set under the objective or real-world probability measure. The variable of which it is necessary to know the current value, or current state, is conveniently referred to as the state variable, here denoted x. The variables chosen at any given point in time are known as the control variables, and here our control variable is π, the weight of the portfolio to be invested in the risky asset. Assuming the state process or variable x is driven by a stochastic process, equation (4.1) is known as a stochastic optimal control problem. We now move on to solving this equation via dynamic programming, importantly in continuous time (that is, by splitting the problem into an infinite number of subsequences), leading to a partial differential equation (PDE) known as the Hamilton-Jacobi-Bellman (HJB) equation. The control problem is then equivalent to finding the solution to the HJB equation.

4.2 Derivation of Hamilton-Jacobi-Bellman (HJB) Equation

A statement of the problem and a general HJB for Lévy processes will be derived before moving on to specializing the equation to an exponential Lévy process with a CRRA investor utility function.

Consider as above the control problem to maximize

EPt,x[U (XTπ)], (4.2)

where the notation

Et,x[·] = E[·|Xt= x].

We consider also the dynamics

dX_tπ = µ(t, X_tπ, π)dt + σ(t, X_tπ, π)dWt+

Z

R

g(t, X_tπ−, z, π) ˜N_t(dt, dz), X₀ = x > 0, (4.3)

with µ(t, x, π) and σ(t, x, π) functions possibly depending on t, x, and π, and g(t, X_tπ, z, π) a function possibly depending on t, x, π, and also the jump size z. The value function is defined by

J (t, x, π) = EP

t,x[U (XTπ)]. (4.4)

The optimal value function is defined by

V (t, x) = sup

π∈Π

{J (t, x, π)}. (4.5)

Now assume two strategies

(21)

Strategy 2. Use the control lawπ? with ? π(s, y) = ( π(s, y), s ∈ [t, t + h] ˆ π(s, y), s ∈ (t + h, T ]. (4.6)

Expected utility for strategy 1. This is trivially J (t, x, ˆπ) = V (t, x) since ˆπ was defined as the optimal control law.

Expected utility for strategy 2. As presented in [7], a stylized interpretation of this strategy is that after falling asleep at time t, you wake up and realize that the state process has moved to point x. You deal as best you can with these circumstances by maximizing your utility over the remaining time, given that you now are starting at time t + h in the state x. So, in the time interval (t + h, T ], since by definition we will use the optimal control law in this time period, we have

J (t, x,π) = E? P

t,x[V (t + h, Xt+hπ )]. (4.7)

Now we’ll compare the two strategies, and since by definition strategy 1 is optimal, we have V (t, x) ≥ EP

t,x[V (t + h, Xt+hπ )]. (4.8)

From Itô’s formula in Theorem 2.9, we have

V (t + h, X_t+hπ ) = V (t, x) + Z t+h t {∂V ∂s(s, X π s) + AπV (s, Xsπ)}ds + Z t+h t ∂V ∂x(s, X π s)σsdWs + Z t+h t Z R {V (s, X_sπ−+ g(s, X_sπ−, z, π)) − V (s, X_sπ−)} ˜N (ds, dz) + Z t+h t Z R {V (s, X_sπ−+ g(s, X_sπ−, z, π)) − V (s, X_sπ−) − g(s, X_sπ−, z, π) ∂V ∂x(s, X π s−)}νs(dz)ds, (4.9) with Aπ = µ(t, X_sπ, π) ∂ ∂x + 1 2σ 2_{(t, X}π s, π) ∂2 ∂x2. (4.10)

Applying the expectation operator, the dWs and ˜N (ds, dz) terms will drop out (as martingales by

definition), and when we plug the result into equation (4.8) we obtain

V (t, x) ≥ EP t,x[V (t, x) + Z t+h t {∂V ∂s(s, X π s) + AπV (s, Xsπ)}ds + Z t+h t Z R {V (s, X_sπ−+ g(s, X_sπ−, z, π)) − V (s, X_sπ−) − g(s, X_sπ−, z, π) ∂V ∂x(s, X π s−)}ν_s(dz)ds] (4.11)

(22)

⇐⇒ EPt,x[ Z t+h t {∂V ∂s(s, X π s) + AπV (s, Xsπ)}ds + Z R {V (s, X_sπ−+ g(s, X_sπ−, z, π)) − V (s, X_sπ−) − g(s, X_sπ−, z, π) ∂V ∂x(s, X π s−)}νs(dz)ds ] ≤ 0. (4.12) Now divide by h, move h within the expectation, and let h tend to zero (so, this is where we incorporate the infinite subproblem, or continuous time assumption). By the fundamental theorem of integral calculus we obtain ∂V ∂t(t, x)+A π_{V (t, x)+} Z R {V (t, x+g(t, x, z, π))−V (t, x)−g(t, x, z, π)∂V ∂x(t, x)}νt(dz) ≤ 0. (4.13) Equality will hold at the optimum, and thus

∂V ∂t(t, x) + sup_π∈Π AπV (t, x) + Z R {V (t, x + g(t, x, z, π)) − V (t, x) − g(t, x, z, π)∂V ∂x(t, x)}νt(dz) = 0. (4.14)

Imposing the additional terminal condition, we have thus arrived at the HJB equation            1) ∂V_∂t(t, x) + sup π∈Π Aπ_{V (t, x)} +R R{V (t, x + g(t, x, z, π)) − V (t, x) − g(t, x, z, π) ∂V ∂x(t, x)}νt(dz) = 0 2) V (T, x) = U (x) ∀x ∈ R. (4.15) 4.3 Admissible Controls

Before specializing the equation and deriving the associated optimal control strategy, it is useful to discuss the details of the set of admissible controls, which we will denote Π. Recall from equation (2.23) that the random jump variable is assumed to have support on [−1, ∞) to guarantee positivity of, or equivalently to preserve the limited liability structure of, the stock price. Furthermore, as discussed in [41], there are additional nuances when considering jump processes in a wealth context. It would seem reasonable to assume that the wealth process itself inherits the behavior of the underlying drivers, that is importantly if all constituent securities are driven by pure diffusions the wealth process will also be a pure diffusion process, however portfolio assets with discontinuous paths would also introduce discontinuities into the overall wealth process. In the case of pure diffusion processes without jumps, the variance of returns over some time period dt is proportional to dt, implying that as one takes dt → 0, the uncertainty associated with changes in wealth also goes to zero. Thus, investors retain complete control over the portfolio via continuous optimal rebalancing, and even leveraged long and short positions can be

(23)

adjusted quickly enough before wealth turns negative. However, this is not the case when discontinuities are present, as the investor may not have enough time to rebalance when a jump occurs, and as a result of this loss of control over the portfolio, negative wealth could arise due to leveraged long and short positions. This situation can be likened to illiquidity risk, since investors in illiquid assets may face large changes in the value of their portfolio before/without being able to rebalance the position. As such, the following proposition applies to the case of jump processes (see [41] or [42] for proof):

Theorem 4.1 (Bounds on Portfolio Weights) For any t, 0 < t ≤ T , and denoting XinfandXsupas the

lower and upper bounds of the support of the random price jumpXt, the optimal portfolio weightπˆt

must satisfy

1 + ˆπtXinf> 0 and 1 + ˆπtXsup> 0. (4.16)

In particular, ifXinf< 0 and Xsup> 0,

− 1 Xsup ≤ ˆπt≤ − 1 Xinf . (4.17)

Considering again our condition for financial applications that the support is [−1, ∞), we find that

−1

∞ ≤ ˆπt≤ − 1

−1 <=> 0 ≤ ˆπ ≤ 1. (4.18)

Thus, the set of admissible controls Π = [0, 1] and the investor will never take leveraged long or short positions in the risky asset.

4.4 Optimal Control of an Exponential Lévy Process Subject to CRRA Utility

Consider the case of iso-elastic or power utility, where

U (x) = (

x1−γ

1−γ for γ > 0, γ 6= 1

log(x) for γ = 1, (4.19)

in a market with two possible investments, 1) A risk-free asset with dynamics

dBt= rtBtdt, B0 = b > 0. (4.20)

2) A risky asset with dynamics

dSt= St−[µ_tdt + σ_tdW_t+

Z ∞

−1

ztN (dt, dz)],˜ S0 = s > 0, (4.21)

with rt> 0, µt> 0, σt> 0 ∈ R variables depending on time. We additionally assume that

Z ∞

−1

(24)

Consider now a wealth or portfolio process given by dPt= Pt πdSt St + (1 − π)dBt Bt , P0 = p. (4.23)

Substituting the asset dynamics from (4.20) and (4.21), we have

dPt= Pt " πSt−[µtdt + σtdWt+ R RztN (dt, dz)]˜ St + (1 − π)rtBtdt Bt # , (4.24)

which can be simplified to the following form dPt= Pt[rt+ (µt− rt)π] | {z } =µ(t,p,π) dt + Z ∞ −1 Ptπzt | {z } =g(t,p,z,π) ˜ N (dt, dz))dt + Ptσtπ | {z } =σ(t,p,π) dWt, (4.25)

also with terminal condition

V (T, p) = U (p). (4.26)

Thus, according to equation (4.15), our HJB equation is, for γ 6= 1            1) ∂V_∂t(t, p) + sup π∈Π Aπ_{V (t, p)} +R∞ −1{V (t, p + pπz) − V (t, p) − pπz∂V∂x(t, p)}νt(dz) = 0 2) V (T, p) = U (p) ∀p ∈ R. (4.27)

The equation for when γ = 1 will be more easily handled at a later stage. We now search for a solution of the form

V (t, p) = U (eδt_p), _(4.28)

where δtis a C1 deterministic function of time such that δT = 0, giving

V (t, p) = (e

δt_p)1−γ

1 − γ for γ > 0, γ 6= 1. (4.29)

Accordingly, note the following ancillary calculations: ∂V ∂t = ˙δt(pe δt₎1−γ_, ∂V ∂p = e δt(1−γ)_p−γ_, ∂ 2_V ∂p2 = γe −δt(1−γ)_p−γ−1_. _(4.30)

To help simplify notation moving forward, let C = ∂V_∂p, and A = ∂_∂p2V2. Realize as well that A can be

rewritten as

A = eδt(1−γ)_p−γ_p−1_{(−γ) = C ·} 1

p· (−γ). (4.31)

Also note that

(25)

As such, consider the full expression for equation 1) of the HJB equation (4.27), ˙δt(peδt)1−γ+ sup π∈Π p[rt+ (µt− rt)π]C + 1 2(pσtπ) 2_A + Z ∞ −1 {e δt(1−γ)_{(p + pzπ)}(1−γ) 1 − γ − eδt(1−γ)_p(1−γ) 1 − γ − e δt(1−γ)_p−γ_pzπ}ν t(dz) = 0 (4.33) ⇐⇒ ˙δt(peδt)1−γ+ sup π∈Π p[rt+ (µt− rt)π]C + 1 2(pσtπ) 2_A + Z ∞ −1 { 1 1 − γ h eδt(1−γ)_{(p + pzπ)}(1−γ)_{− e}δt(1−γ)_p(1−γ)i_{− e}δt(1−γ)_p(1−γ)_zπ}ν t(dz) = 0. (4.34) Notice now we can factor Cp = eδt(1−γ)_p(1−γ) _{out of each term, resulting in the final form of our HJB}

equation when γ > 0, γ 6= 1,            1) ˙δt(peδt)1−γ+ eδt(1−γ)p(1−γ)sup π∈Π rt+ (µt− rt)π −1₂γσ2tπ2 +R₋₁∞{_1−γ1 (1 + zπ)(1−γ)_{− 1 − zπ}ν} t(dz) = 0 2) V (T, p) = U (p) ∀p ∈ R, (4.35)

but due to the strict positivity of (peδt₎(1−γ)_{and the fact that V (T, p) = U (e}δT_{p) = U (p) =⇒ δ}

T = 0, this is equivalent to            1) ˙δt+ sup π∈Π rt+ (µt− rt)π −1₂γσ2tπ2 +R∞ −1{1−γ1 (1 + zπ) (1−γ)_{− 1 − zπ}ν} t(dz) = 0 2) δT = 0. (4.36)

Now, we also revisit the limiting case when γ = 1, invoking the Cauchy/L’Hôpital theorems, and find that            1) ˙δt+ sup π∈Π rt+ (µt− rt)π −1₂γσt2π2 +R₋₁∞{[log(1 + zπ) − zπ] νt(dz) = 0 2) δT = 0. (4.37)

Remark (Time-Invariance of Optimal Strategy) Since neither equation 1) of the above HJB equations depends on the state variablesP , S, or the time to maturity T − t, the optimal strategy will be totally myopic or time-invariant in the sense that the variableπ controlling the proportion of wealth invested inˆ the risky asset will not depend on the current level of wealth, but only on the variablesµt,rt,σt, and the

(26)

4.5 Existence of Solution

Consider the first equation of the general HJB given in equation (4.15) and its full form, ∂V ∂t(t, x) + sup_π∈Π [µ(t, x, π) ∂ ∂x+ 1 2σ 2_{(t, x, π)} ∂2 ∂x2]V (t, x) + Z ∞ −1 {V (t, x + g(t, x, z, π)) − V (t, x) − g(t, x, z, π)∂V ∂x(t, x)}νt(dz) = 0. (4.38) Let LπV (t, x) = [µ(t, x, π) ∂ ∂x+ 1 2σ 2_{(t, x, π)} ∂2 ∂x2]V (t, x) + Z ∞ −1 {V (t, x + g(t, x, z, π)) − V (t, x) − g(t, x, z, π)∂V ∂x(t, x)}νt(dz). (4.39) Lπ_{V (t, x) is called the infinitesimal generator of the controlled process X}π_{, which are linked by the}

Dynkin formula, EP[f (T, XTπ)] − EP[f (t, Xtπ)] = EP Z T t Lπu_{f (u, X}π u)du , (4.40)

for f good enough. We now state the verification theorem for exponential additive processes as given in [58]:

Theorem 4.2 (Verification Theorem) Define the set D by

D := {f ∈ C1,2_{([0, T ] × R)|∀t ∈ [0, T ], (4.40) holds ∀π ∈ Π}.} (4.41)

LetK ∈ D be a classical solution to the HJB equation (4.15). Then for all (t, x) ∈ [0, T ] × R, 1)K(t, x) ≥ J (t, x, π) for every admissible controlπ ∈ Π;

2) there exists an admissible controlπ ∈ Π such thatˆ ˆ

π(s, x) ∈ arg max

π L

π_{K(s, x) ∀ (s, x),}

thenK(t, x) = J (t, x, ˆπ) = V (t, x).

Consider now that equations (4.36-4.37) are of the type ˙

δt+ λt= 0, (4.42)

still with terminal condition δT = 0, thus yielding δt=

RT

t λudu and a candidate value function of

V (t, p) = U (peRtTλudu) = (pe

RT

t λudu)1−γ

(27)

with a candidate optimal control ˆπ. In order to apply Theorem 4.2, we need to prove that V ∈ D. Applying Itô’s formula to the candidate value function (see Theorem 2.9 and equations (4.9), (4.25) and (4.30) for guidance here), we have, for γ 6= 1

dV (t, P_tπ) = ∂ ∂t( Z T t λudu) P_tπe RT t λudu (1−γ) dt + P_tπ[rt+ (µt− rt)ˆπ] eRtTλudu (1−γ) (P_tπ)−γdt − 1 2γP 2 tσt2πˆ2 eRtTλudu (1−γ) (P_tπ)(−γ−1)dt + e RT t λudu (1−γ) (P_tπ)−γP_tπσtπdWˆ t + Z ∞ −1 {e δ(t)(1−γ)_(Pπ t + Ptπzπ) (1−γ) 1 − γ − eRtTλudu(1−γ)(Pπ t) (1−γ) 1 − γ } ˜N (dt, dz) + Z ∞ −1 {e RT t λudu(1−γ)(Pπ t + Ptπzπ) (1−γ) 1 − γ − eRtTλudu(1−γ)(Pπ t) (1−γ) 1 − γ −e RT t λudu(1−γ)(Pπ t) −γ P_tπzπ}νt(dz). (4.44) Recalling equation (4.32), note that we can factor

eRtTλudu

(1−γ)

(P_tπ)(1−γ) out of every term. Also, recall that by the fundamental theorem of calculus, we have that _∂t∂(R_tT λudu) = −λt. Thus, we can

simplify to dV (t, P_tπ) = P_tπe RT t λudu (1−γ) −λt+ rt+ (µt− rt)ˆπ − 1 2γσ 2 tπˆ2 + Z ∞ −1 { 1 1 − γ[(1 + z ˆπ) (1−γ)_{− 1] − z ˆ}_π}ν t(dz) dt +P_tπe RT t λudu (1−γ) σtπdWˆ t + P_tπe RT t λudu (1−γ)Z ∞ −1 [(1 + ˆπz)(1−γ)− 1] ˜N (dt, dz). (4.45)

Noting that the dt term is LπV (t, P_tπ) and further denoting the remaining terms, as in [58], as dMt, we

can thus express this as

dV (t, P_tπ) = LπV (t, P_tπ)dt + dMt. (4.46)

Note that by analogous logic, one finds that in the case of γ = 1, dMt= ˆπσtdWt+

Z ∞

−1

log(1 + ˆπz) ˜N (dt, dz). (4.47)

For γ > 0, the Dynkin formula of equation (4.40) thus holds if EP[

Z T

t

dMt] = 0, (4.48)

that is, if M is a martingale. As the sum of a stochastic integral and a compensated Poisson integral, this condition is clearly satisfied. Thus, we can apply the verification theorem, and we now turn the focus to the optimal strategy.

(28)

4.6 Optimal Strategy

First, it is important to note that the existence result above holds only over the interior of the compact set Π = [0, 1]. However, solutions on the boundary are possible and must be evaluated case by case. Nevertheless, we can analyze the first order condition and derive a more explicit solution by assuming an internal solution. We are thus interested in, for γ 6= 1,

sup π∈Π rt+ (µt− rt)π − 1 2γσ 2 tπ2 + Z ∞ −1 { 1 1 − γ h (1 + zπ)(1−γ)− 1i− zπ}νt(dz) , (4.49) and for γ = 1, sup π∈Π rt+ (µt− rt)π − 1 2γσ 2 tπ2 + Z ∞ −1 {[log(1 + zπ) − zπ] νt(dz) . (4.50)

Working through the first order condition_∂π∂ = 0 for both, recalling for the integral term the Leibniz rule

d dx Rb af (x, t)dt =Rb a ∂ ∂xf (x, t)dt, we obtain (µt− rt) − γσ2tˆπ + Z ∞ −1 z (1 + z ˆπ)γ − z νt(dz) = 0 (4.51) ⇐⇒ (µt− rt) − γσt2π +ˆ Z ∞ −1 z((1 + z ˆπ)−γ − 1)νt(dz) = 0, (4.52)

for all γ > 0. It is obvious from the equation that no explicit expression exists for the control ˆπ, however numerical methods provide a simple solution.

5 Transform Methods in Option Pricing

5.1 The Carr-Madan Method

We now move on to discussing option pricing by the Fourier transform and the efficient Fast Fourier Transform algorithm. Carr and Madan [13] lay the framework for using the Fourier transform to solve the option pricing problem. It should be noted that this method is only technically valid for options of European expiry style, though a popular related technique known as the convolution or CONV method, which can accommodate early exercise features, can be found in [43]. Furthermore, the reader will note a few points. First, this section restricts the formulations to the case of near-the-money (vanilla) call options, as this is all the later empirical study will utilize. This does not, however, mean that this is the extent of the capability of the framework. First, to obtain put prices, one can either simply use put-call parity, or utilize a closed-formulation for the put price, which is easily obtained by applying some obvious analogous calculus as will be presented for the case of call options, or referring to a multitude of easy to find sources. Furthermore, Carr and Madan also present an important extension to the case of out-of-the-money options near maturity, as some numerical difficulties arise in these cases when using the base methodology. Though very useful, since the empirical study only considers near-the-money options with at least thirty days to maturity (more on the specifics of the data used later), we will skip the details of this alternate pricing formula and the reader should refer to [13] if needed.

(29)

5.1.1 The Fourier Transform of a Near-the-Money Vanilla Call Option

Consider the classical case of pricing a European call option. Risk neutral valuation yields CT(k) = e−rTEQ[max(ST − K, 0)] = e−rT

Z ∞

k

(es− ek_)q

T(s)ds, (5.1)

where CT is the time 0 price of the claim maturing at time T , s = log(S), k = log(K), and qT is the

risk-neutral density of sT. However, one encounters an issue as k → −∞, as CT(k) approaches the

underlying value, CT(k) → S0, and is thus not square-integrable. That is, consider equation (5.1) with

k = −∞, resulting in CT(−∞) = e−rT Z ∞ −∞ (es− e−∞)qT(s)ds = e−rT Z ∞ −∞ esds =⇒ Z ∞ −∞ |es|2ds = ∞. (5.2)

Accordingly, Carr and Madan proposed using a modified call price to remedy this problem, by introducing a "dampening effect," multiplying the call price by eαkand effectively inducing exponential decay as k becomes negative. An illustration of this effect, provided in [33], can be seen below to develop the intuition.

With the modified call price satisfying the square-integrability condition, consider now the Fourier transform of cT(k) = eαkCT(k),

ΨT(ν) =

Z ∞

−∞

eiνkcT(k)dk. (5.3)

The square-integrability importantly allows for the inversion of this transform, yielding, for the call price, cT(k) = 1 2π Z ∞ −∞ e−iνkΨT(ν)dν (5.4) = CT(k) = e−αk 2π Z ∞ −∞ e−iνkΨT(ν)dν, (5.5)

which can be expressed as

CT(k) = e−αk π Re[ Z ∞ 0 e−iνkΨT(ν)dν], (5.6)

(30)

due to the fact that call option prices must be real values, and thus ΨT(ν) is odd in its imaginary part and

even in its real part. An analytical expression for ΨT(ν) in terms of φT(ν), the characteristic function of

the log return, exists and is given by (for Proof see Appendix E): ΨT(ν) =

e−rTφT(ν − (α + 1)i)

α2_{+ α − ν}2_{+ i(2α + 1)ν}. (5.7)

Substituting this result for ΨT(ν) in the pricing formula (5.6) and evaluating the expression will yield

the price of the call option as: CT(k) = e−αke−rT π Re[ Z ∞ 0 e−iνkφT(ν − (α + 1)i) α2_{+ α − ν}2_{+ i(2α + 1)ν}]. (5.8)

The remaining problem is therefore how one should evaluate equation (5.8). As the option price has been expressed as a direct Fourier transform, the Fast Fourier Transform (hereafter FFT) algorithm is well suited.

5.1.2 The FFT Algorithm for Vanilla Option Pricing

The FFT algorithm is a method for computing sums of the form

N −1

X

j=0

e−i2πNjkx_j, k = 0, ..., N − 1. _(5.9)

We can approximate CT(k) by applying the trapezoidal integration rule and setting νj = η(j − 1),

resulting in CT(k) ≈ e−αk π N −1 X j=0 e−iνjk_Ψ T(νj)η, (5.10)

where η > 0 is the integration step, νj = η · j, and k is a vector of strikes near the money, ranging from

θ−b to θ+b with θ = ln(S0), b = π_η and N steps of size λ = _{N −1}2b , or k = θ−b+λu, u = 0, ..., N −1.

To obtain an accurate integration with larger values of η, Carr and Madan suggest incorporating the Simpson weighting rule, resulting in

CT(k) ≈ e−αk π N −1 X j=0 e−iνjk_Ψ T(νj)η · Sw (5.11) ≈ e −αk π N −1 X j=0 e−iνjk_Ψ T(νj) η 3(3 + (−1) j+1_{− δ} j), (5.12)

with δj the Kronecker delta function. Substituting νjand k into the above equation yields

CT(k) ≈ e−αk π N −1 X j=0 e−iηj(θ−b+λu)ΨT(ηj) η 3(3 + (−1) j+1_{− δ} j). (5.13)

(31)

Simplifying a little yields the following final form of the call price formula: CT(k) ≈ e−αk π N −1 X j=0

e−iηjλuΨT(ηj)eiηj(b−θ)

η 3(3 + (−1) j+1_{− δ} j) | {z } =xj . (5.14)

Recall that the FFT algorithm computes sums of the form given in equation (5.9), where we see that for our application we need ηλ = 2π_N. As long as N is chosen as a power of 2 (at least 214is a good rule of thumb), and all other parameters are set as defined above, this condition will be satisfied.

5.2 Lévy Process Parameter Calibration Using Transform Pricing

Taking market prices as a reference, the transform pricing method presented above can be used to calibrate the parameters of a given Lévy market model. That is, the option pricing problem is inverted, with market prices assumed as given, and the ETS process calibrated as to most closely replicate these given data points. To this end, an objective or cost function is constructed to achieve the best approximation to the data. A natural choice for a convex objective function is of the least squares form, given by

N LS = min ˜ − →_{v ,˜}_η 1 N N X i=1

[(market price)i− (calculated price)i]2, (5.15)

for each i strike and where a tilde represents the parameter under the risk-neutral measure. Note that η here is from the ETS model, and this differs from the η of the FFT algorithm as presented. However, as noted in [44], this results in a bias towards more expensive in-the-money call options over the relatively cheaper out-of-the-money options. Thus, as proposed in for instance [2], we will use an average pricing error, given by AP E = min ˜ − →_{v ,˜}_η 1 N N X i=1

market price_i− calculated price_i market price_i . (5.16)

A significant issue remains due to this problem being ill-posed, in the sense that the calibration may be non-unique and highly sensitive to the initial value choices. There are a few ways we can deal with this. The first and perhaps most intuitive, though heuristic, is to simply run the optimization multiple times with different initial value estimates, and choose the eventual calibration which minimizes the cost function of equation (5.16). Alternatively, we can utilize the regularization framework developed in [18], [19], and [20], which draws heavily on the minimal entropy measure concepts of sections 2 and 3. This regularization framework is detailed in Appendix K. Both methods (that is, the multiple initial values approach and a regularized approach) were tested as part of the empirical study to be described in the next section, and the method of simply using multiple initial value estimates for the parameter vector and minimizing equation (5.16) was found to outperform the regularized approach. Accordingly, the reader should assume the use of the multiple initial value approach whenever calibration is discussed moving forward, and as mentioned can find a more detailed analysis of the findings in Appendix K.

(32)

6 Data, Methodology, and Results

6.1 Dataset

The dataset for the empirical analysis covers the period of January 2004 - December 2017. A set of daily option chains on the Standard and Poor’s 500 index (S&P 500) was obtained from the Chicago Board Options Exchange (CBOE), known as the SPX Options Traditional product. The CBOE data has the benefit of including a quote at 3:45 PM U.S. Eastern time (15 minutes before the market close). The contract has a European exercise style and is AM-settled on the 3rd Friday of every month. Only call options were considered in the calibration of the ETS model. The average of the market bid and ask was used for the market price, and following [61] and [23], the following filters were imposed upon the data: 1) Moneyness is constrained to −0.10 ≤ (K/St− 1) ≤ 0.10,

2) Annualized implied volatility must be between 5% and 95%,

3) Remove options with premia below the lower bound, in particular max(0, St− qtSt− Ke−rt(T −t))

for call options,

4) Verify that call prices are a decreasing function of strike.

However, the option data was used in another important way, and in this case these filters proved far too restrictive, so it may be a good point to detail this consideration. In order to add context to the performance of the strategy, we of course need a suitable benchmark. In the spirit of using option-implied information to guide allocation decisions, the main benchmark will be a mean-variance version of the stochastic optimal control framework of section 4, where variance is implied from option prices using the well-known "model-free" approach (see [11],[30], and [14]). This represents an interesting benchmark because, while similarly relying on option-implied information, it also contrasts the ETS model in a number of ways. First, it is of course a mean-variance type optimization, and therefore it allows us to differentiate the performance of a portfolio that is incorporating the option market’s expectations of the higher moments of the distribution of forward returns from a portfolio that does not. Secondly, as stated, it is a non-parametric or model-free approach and therefore does not rely on any option pricing model. Again, the filters listed above are far too restrictive for this methodology, and produce implied variance numbers that are nonsensically low. Accordingly, model-free implied variance is calculated in similar fashion to the widely followed VIX index (see [15] and [62]) as follows. First, the options used encompass a cross section of only out-of-the-money options, meaning that the options used are puts when St< K and calls when St> K. No limit is initially placed on how "deep" out-of-the-money

these options are allowed to be, however no zero bid options are included, and the data set is cut off when instances arise of two consecutive zero bid strikes. As before, option prices are taken as the average of the bid and ask price and have 30 calendar days to maturity. Model-free implied variance is then computed as σ_{M F IV}2 = 2 t X i 4Ki K2 i ertQ(Ki), (6.1) where

(33)

4K_i = distance between successive strikes.

Q(Ki) = mid-point of bid-ask spread for an option with strike Ki.

The optimal control in this framework is the the well-known result of [51] ˆ

π = µt− rt

γσ_{M F IV}2 . (6.2)

Two benchmark portfolios will actually be formed based on this mean-variance optimal control. Recall from Theorem 4.1 that under the assumption that the risky asset (and by association the wealth process) can experience discontinuities, the investor will never take levered long or short positions in the risky asset. However, as also explained in section 4, in the case of pure diffusion processes, the investor retains a level of control over the portfolio that admits levered long and short positions. Accordingly, a non-levered version of the mean-variance portfolio, denoted "MFIV", which restricts the weights produced from equation (6.2) above to the same set of admissible controls for the ETS portfolio (ˆπ ∈ [0, 1]) serves as one benchmark. Then, there is a second levered benchmark, denoted "MFIVHL", where "HL" stands for half-leverage, signifying that the leverage is tempered by the popular rule-of-thumb in practice for managing the risk of levered positions (i.e., the "half-Kelly" rule when considering log-utility). This helps temper the leverage to more realistic levels, but note this is only one such way to do so. In for instance [37], leverage was constrained by restricting the optimal control to the set [−1, 2]. Additionally, the underlying spot index price along with the associated index dividend yield as well as the risk-free interest rate, chosen to be one-month LIBOR, were obtained from Bloomberg. For the investable strategy to be detailed in the next section, daily adjusted close prices for the SPDR S&P 500 ETF (ticker SPY) were also obtained from Bloomberg. Finally, a result in the next section will show that when changing from the risk-neutral to the real-world parameterization of the Lévy density, we need a number for the expected market return, ¯µ. In order to keep with the theme of using forward looking implied information but also ensuring to always satisfy the second condition of equation (4.22), this is taken to be a long-term implied market risk-premium for the US market, available from http://www.market-risk-premia.com/us.html. For some transparency and for the reader’s convenience, the calculation methodology is recreated in Appendix F.

6.2 Methodology of Investment Strategy

The first five sections of the paper lay the theoretical groundwork and supply all the tools needed for the following analysis. The goal is to take these concepts and investigate if they can be holistically employed in practice to improve the allocation choice problem and ultimately portfolio performance. As discussed in the introduction, there have already been studies, for instance [37] as mentioned but also [21], [26], and [27], which show that option-implied information can provide valuable signals in constructing portfolios. Further, in for instance [23] it was found that market information extracted under an exponential Lévy market assumption (although here only the Meixner and Normal Inverse Gaussian models were considered) showed that the risk preferences of equity investors were signaling anomalous

Dynamic asset allocation using option implied distributions in an exponentially tempered stable Lévy market

M

ASTER

M

ATHEMATICAL

F

INANCE

M

ASTER’S

F

INAL

W

ORK

D

D

A

A

U

O

I

D

I

A

E

T

S

L

M

M

ASTER

M

ATHEMATICAL

F

INANCE

M

ASTER’S

F

INAL

W

ORK

D

D

A

A

U

O

I

D

I

A

E

T

S

L

M

Z

P

S

J

G

M

G

S

2018

Dynamic Asset Allocation Using Option-Implied Distributions in

an Exponentially Tempered Stable Lévy Market

Contents

1

Introduction

2

Lévy Processes

χ

χ

χ

Π

3

The Exponentially Tempered Stable (ETS) Processes

χ

χ

4